Discontinuous Finite-Element Galerkin Method for Numerical Solution of Parabolic Problems in Anisotropic Media on Triangle Grids

Abstract

A new numerical algorithm for solving parabolic initial-boundary values problems in anisotropic media is proposed. The algorithm is based on Galerkin method with discontinuous basic functions on triangle meshes. The 2nd order derivatives can't be directly harmonized in a weak variational formulation using the discontinuous functions' space. Hence additional variables are introduced to reduce the initial 2nd-order equation to the system of the 1st-order equations. The special feature of this method is in consideration of additional variables within a dual mesh. The dual mesh consists of median control values and is conjugate to the initial triangle mesh. The stream values on the element boundaries are calculated with addition of stabilizing additives. The method is studied basing on the example of 2-dimensional parabolic boundary problems. Convergence and accuracy of the method are investigated. Calculations in model problem show the possibility to use the method discussed for solving parabolic problems in anisotropic media on triangle meshes.